# Quantum mechanics: bound state

I have a one dimensional quantum mechanical system composed by a particle in a potential $$V(x) = -\frac{1}{ma^2} \textrm{sech}^2(x/a)$$ The problem gives me the eigenstates $$\psi_k(x) = \frac{ika -\tanh(x/a)}{ika+1} \frac{e^{ikx}}{2 \pi}$$ so that the eigenvalues are $$E_k = k^2/2m$$. Now I have to compute the integral $$\int_{-\infty}^{\infty} dk \langle x| \psi_k \rangle \langle y|\psi_k\rangle$$ and the problem asks me if the $$\psi_k$$ with $$k \in \mathbb{R}$$ form a basis of the Hilbert space.

I have tried to compute the integral with the residue theorem. The integral splits in four parts, one of these part is $$\int_{-\infty}^{\infty} \frac{k^2 a^2}{k^2a^2+1} e^{ik(x-y)} dk$$ but this integral is not convergent because the asymptotic behavior for very large values of $$k$$ is $$\sim e^{ik(x-y)}$$ and so the integrand does not decrease to zero.

Now the problem asks me to prove that there is only one bound state $$\psi_0$$ and to determine $$\psi_0$$ and his eigenvalue. The problem gives me also an hint

Use the result of the integral $$\int_{-\infty}^{\infty} dk \langle x| \psi_k \rangle \langle y|\psi_k\rangle$$ to guess the form of $$\psi_0$$.

But as I write above, that integral seems divergent!

If you add $$\pm 1$$ in the numerator you get a Dirac delta plus other pieces for which you can use residue theorem. Indeed: $$\int_{-\infty}^{+\infty}dk \frac{k^2a^2}{k^2a^2+1}e^{ik(x-y)}= 2\pi\delta(x-y)-\int_{-\infty}^{+\infty}dk \frac{1}{k^2a^2+1}e^{ik(x-y)}$$. Let $$I$$ be the last integral. The integrand function has two poles at $$k=\pm i/a$$ and it is $$\mathcal{O}(k^{-2})$$ for large $$k$$, so you can use residue theorem to evaluate $$I$$. Choosing the proper contour w.r.t the sign of $$(x-y)$$, you can verify that (if I'm not wrong) $$I=\pi a e^{-|x-y|/a} \, .$$
• Thank you for your answer. The result of the complete integral is: $\delta (x-y) + \frac{1}{2a} e^{-\frac{x-y}{a}}(\textrm{tanh}(x/a)\textrm{tanh}(y/a) + \textrm{tanh}(y/a) - \textrm{tanh}(x/a) -1)$. If the basis was complete the result would have been only the Dirac delta. So I imagine I should write something like: $\int_{-\infty}^{\infty} dk \langle x| \psi_k \rangle \langle y|\psi_k\rangle + \langle x | \psi_0 \rangle \langle y| \psi_0 \rangle$ and, forcing this expression be equal to Dirac delta, find the form of $\psi_0$. Is this the right way? – dfgoe55 Feb 29 '20 at 15:49
• Since it is a one dimensional problem with only a bound state, the wave function shouldn't have any node. I don't figure out how the previous result of the integral helps me to guess the form of $\psi_0$. Yes,it is a SISSA problem... – dfgoe55 Feb 29 '20 at 16:45